A new approach to marking volatility of illiquid options

Valer Zetocha, head of quantitative services for equity and foreign exchange at Julius Baer, calls it an “old” problem, but a “very important one”.

Liquidity in options that reference rarely traded assets is bound to vary across maturities. Marking the volatility of those maturities is challenging, making it difficult to build a volatility surface that is arbitrage-free and consistent with market pricing.

Over the years, many solutions have been proposed – ranging from the use of implied volatility models to the estimation of underlying price distributions for different maturities. Each approach has numerous variations, with their own advantages and disadvantages, but none work well across all maturity, moneyness and liquidity levels.

Zetocha proposes a new approach, which helps existing models to output a smooth and workable volatility surface when dealing with illiquid maturities.

The first step is to organise the maturities in order of decreasing liquidity – either based on the average spread or the grid density of strikes quoted in the market – rather than chronologically. This is key to maintaining the stability of the volatility surface. Once sorted by liquidity, the volatility surface of the most liquid maturity is built first, followed by the next, and so on. While following this bootstrapping process, Zetocha deploys a part of the algorithm he calls the ‘arbitrage sniper’ to eliminate outliers and arbitrageable quotes.

It might be surprising to learn that arbitrageable prices still need to be eliminated, considering bid-offer spreads should prevent them. But when dealing with mid-prices, some circumstances can theoretically create arbitrage, such as bad data or inconsistent information from market-makers.

The algorithm then uses the entropy maximisation technique presented by Neri and Schneider in 2012 to construct the full skew of the maturity, such that it satisfies the constraints of the options that are already quoted.

Time translation

At this stage, Zetocha introduces a formula to time-translate the probability distribution obtained using the Neri-Schneider technique and diffuses it into longer-maturity distributions. This preserves the characteristics of the original distribution and is consistent with the empirical observation that the distribution flattens as time progresses and the stochastic variable diffuses. This way, the volatility distribution can be moved from one maturity to another.

“The introduction of the time-translation formula is, in my opinion, the most valuable contribution of the paper,” says Zetocha. “It’s a very generic algorithm. It can translate the probability distribution function from one maturity to another in a way that guarantees that you don’t produce arbitrage, and this can be done for any asset class.”

Zetocha’s objective is not to propose a new parametrisation or approach to represent volatility data. Rather, the paper complements existing frameworks and can be plugged into existing pricing libraries. The solution aims to fill in the sparse volatility surface that is typical of illiquid stocks and synthesise the volatility for strikes and maturities that are non-existent in the market.

“Once you have that in a non-arbitrageable way, any parametrisation, including the simplistic splines, has a good chance of working,” says Zetocha. He adds that the approach “is very time-efficient”, taking less than 200 milliseconds in most cases. “The only condition for applicability is that the asset can be modelled as a stochastic process.”

Zetocha says the algorithm is applicable to any asset class, and can generate curve values from as little as just one maturity.

Experienced traders have a knack for spotting outliers and arbitrageable prices and exploiting them. Zetocha’s approach can automate that process and allow firms to build quicker volatility surfaces.